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直线和圆的位置关系典型习题
1.如图,AB为⊙O的直径,C为⊙O上一点,AD和过点C的切线垂直于D,BE和过点C的切线垂直于E求证
(1)AC平分∠DAO
(2)BC平分∠EBO
(3)DC=CE
(4)AD+BE=AB
(5)△ADC∽△ACB
(6)△CEB∽△ACB
(7)AC2=AD·AB
(8)BC2=BE·AB
2.已知PA、PB分别切⊙O于A、B,C为优弧上一点,D为劣弧上一点求证
(1)∠D=90°+∠P
(2)∠C=90°-∠P
3.如图,两个同心圆,大圆的弦AB和小圆相切,AB=a求证圆环的面积=
4.已知AB为⊙O直径,AD∥BC,∠B=90°,DC切⊙O于E求证
(1)CD=AD+BC
(2)∠COD=90°
(3)OE2=AD·BC
(4)连结AC、BD,交于点F,则EF∥BC
(5)延长EF交AB于G,则GF=EF
5.Rt△ABC中,∠B=90°,AB=6,BC=8,则△ABC的内切圆半径为___________,△ABC的外接圆半径为___________
6.已知PA、PB分别切⊙O于A、B,E为劣弧AB上一点,过E点的切线交PA于C、交PB于D
(1)若PA=6,求△PCD的周长
(2)若∠P=50°求∠DOC
7.已知圆外切四边形ABCD,则AB︰BC︰CD︰DA=6︰4︰5︰_____(已知圆内接四边形ABCD,则∠A︰∠B︰∠C︰∠D=2︰3︰4︰__)
8.已知△ABC中,AB=AC=13,△ABC的面积为60,求△ABC内切圆直径(此题两个结果,可用相似求,也可利用S△=C△·r)
9.已知如图,PA、PB分别切⊙O于A、B,AC为⊙O直径求证
(1)PO∥BC
(2)△ABC∽△PBO
(3)PO·BC=2OB2
(4)
(5)E为△PAB的内心
10.如右图,点I是△ABC的内心,AI交边BC于点D,交△ABC的外接圆于点E求证
(1)BE=IE
(2)IE2=DE·AE
11、等边三角形的边长为,则它的内切圆半径为______,外接圆半径为______,高线为_______
12、在△ABC中,∠A的平分线AE交BC于D,交△ABC的外接圆于E
(1)连结BE、CE,图中有哪些相等的角?有哪些相等的线段?并证明
①△ADC∽△BDE∽△ABE
②△ABD∽△CED∽△AEC
(2)过E作圆的切线MN,交AB的延长线于M,交AC的延长线于N,证明MN∥BC
(3)过B作圆的切线BF,证明BE平分∠DBF
(4)过A作圆的切线AG,交BC的延长线于G,证明△AGD为等腰三角形
123413、已知AD切△ABC的外接圆于A,交BC的延长线于D,DF平分∠ADB,分别交AC、AB于E、F求证
(1)DC︰DB=AC2︰AB2
(2)AE=AF
(3)图中共有多少组相似三角形
(4)AE2=CE·BE
14、过⊙O内部或外部的一点P作⊙O的割线,交⊙O于A、B两点求证PA·PB=∣OP2-r2∣(r为⊙O的半径)
15、半径为5的⊙O中,半径OC垂直于直径AB,割线PC交⊙O于D,交AB的延长线于P,若CD=2,分别求PC、PB、BD的长(两种方法
①列方程,
②作OE⊥PC于E)
16、已知AB为⊙O直径,∠BAC=20°,D为的中点,ED切⊙O于D,求∠EDA
17、延长⊙O的直径AB到P,使PB=OB,过P作PC切⊙O于C,过B作BD切⊙O于B,交PC于D
(1)求∠P、∠A的度数
(2)求CD︰DP
18.已知AB是⊙O的直径,BC是⊙O的切线,切点为B,OC∥AD,
(1)求证DC是⊙O的切线,
(2)若⊙O的半径为R,求证AD×OC=2
(3)若BC=6,⊙O的半径为
3.求AD的长
(4)设OC交⊙O于N,当AE AO等于多少时,四边形AOND菱形,请说明理由
19.如图△ABC中,AB=AC,以AB为直径的⊙O交BC于D,DE⊥AC于E,交BA的延长线于H
(1)求证DE是⊙O的切线,
(2)若AH=,tan∠FBD=求⊙O的直径
(3)试探究
①四边形AEFG的形状
②BD,BA,BF三条线段之间的关系式,
③过H作∠DHB的平分线交BC于M,当AB的长度变化时H在直线AB上的位置也发生变化,试问∠HMD的大小是否发生变化,若不发生变化,求出∠HMD的度数;若发生变化,请说明理由
(4)AH=,tan∠FBD=求BG的长度.
20.如图以Rt△ABC的直角边AC为直径的⊙O交斜边AB于D,过O作OE∥AB,交BC于E
(1)求证DE是⊙O的切线,
(2)CE=BE
(3)如果⊙O的半径为,ED=2求AB的长
(4)当∠BAC为多少度时,AOED为平行四边形?
(5)在
(4)的条件下,探索CODE的形状
21.如图AB=AC∠ABC=以AC为直径作⊙O,CO与⊙O交于D,AD交BC于E,F是BE的中点,
(1)判断DF与⊙O的位置关系
(2)下列四个结论
①=CE×CB
②4=ED×EA
③∠OCB=∠EAB
④DF=CD其中正确结论是(只填序号)并选择一个证明
22.已知点A,B,C,D在⊙O上,DA=DC,AC,BD相交于E,DE=EB,延长BA至F,使AF=AB,连接DF
(1)BD平分∠ABC
(2)若DE=3,求AB的长{3}试判断DF与⊙O的位置关系
(4)求∠F的正弦值
(5)探索ABCD与BODC的形状
23、如图AB是⊙O的直径,AC=CE,CD⊥AB于D,AE交CD于F,交BC于K
(1)求证AF=CF
(2)若tan∠ECA=CD=2,求⊙O的直径
(3)连接OF,试判断OF与BK之间的关系
(4)写出图中至少三个不同的比例式或乘积式,并选择一个证明
24.在△ABC中AB=AC,O为AB的中点,以O为圆心,OB为半径的圆交BC于D,过D作DE⊥AC于E
(1)求证DE是⊙O的切线
(2)若点O沿AB向点B运动,其它条件不变,那么DE与⊙O有什么位置关系?并证明
(3)若AB=AC=5,SinA=,那么圆心O在AB的什么位置时,⊙O与AC相切?请说明理由8知△ABC内接于以AB为直径的⊙O过C作⊙O的切线交AB的延长线于D,DA AB=12
(1)求∠CDB的度数
(2)在切线DC上截取CE=CD,连接EB,判断EB与⊙O有什么位置关系?并证明
(3)利用图中已标明的字母,连接线段,指出至少5对相似三角形(不包含全等,不需要证明)25.如图A为⊙O的直径EF上的一点,OB是和这条直径垂直的半径,BA和⊙O相交于另一点C,过C的切线交EF的延长线于D,
(1)求证
①DA=DC
②AbxAC=AExAF2当DF EF=18且DF=时,求ABxAC的值
(3)现将直径EF所在的直线向上平移到如图所示的位置其它条件不变,试猜想DA=DF是否仍然成立?
26、如图1,ABAC分别为⊙o的直径和弦,D为EQ\o\acBC的中点,过点D的切线与AC的延长线相交于点E⑴求证DE⊥AC;⑵求证AD2=AE·AB;⑶试判断EAECED三条线段之间的数量关系,并证明;⑷过点D作DF⊥AB于F,如图2,求证EC·EA﹦AF·BF;
27、如图1,已知AB是⊙o的直径,延长AB到D,过D作⊙o的切线DC,C为切点⑴求证DC2=DA·DB;⑵如图2,求证=;⑶若DC=6,=,求AB的长;⑷如图3,过点C作CG⊥AB于F,交⊙o于G,若=,BD=9求⊙o的半径及SinD;
28、已知AB是⊙o的直径,BC是切线,切点为B,DC是⊙o的切线⑴求证OC∥AD;⑵若圆的半径为R求证AD·CO﹦2R2;⑶在⑵的条件下,若OA﹦2,AD﹢OC﹦6,求CD的长;⑷延长CD,BA相交于点E,若EA﹦1,ED﹦2,求tan∠EDA;⑸若AD﹦2DC﹦,求tan∠EDA;⑹若AE﹦3CD:DE﹦23,求⊙o的半径;⑺若BC﹦6⊙o的半径为3,求AD的长;⑻若AB﹦BC﹦2求ED:EC;
29、如图,在△ABC中,以BC为直径的⊙O交AB于点F,E为AC上一点,EF是⊙O的切线,E为AC的中点﹙1﹚求证AC是⊙O的切线;﹙2﹚求证AF·AB=4EF2;﹙3﹚连接OE,若四边形OEFB是平行四边形,求Sin∠ABE的值;
30、△ABC是⊙o的内接三角形,弦AD平分∠BAC,交BC于点E⑴求证AB·AC﹦AD·AE;⑵求证BD2﹦DE·DA;⑶求证AD2﹣BD2﹦AB·AC;⑷求证AE2﹢BE·EC﹦AB·AC;
31、如图,四边形ABCD内接于⊙o,过点C的直线分别交AB、AD于F、H,AC平分∠BAD,HF∥BD⑴求证FH是⊙o的切线;⑵求证BC2﹦CE·CA;⑶求证CD2﹦AB·DH;⑷若I是△ABD的内心,试判断CI、CE、CA之间的数量关系,并证明;32﹒AB﹦AC以AB为直径⊙O交BC于点D,交AC于点E﹙1﹚求证DE﹦BC﹙2﹚已知AB﹦5cm,DE﹦3cm,求BE的长﹙3﹚过点D作DF⊥AC求证F是CE的中点﹙4﹚在⑶的条件下,求证BC2﹦4CF·CA﹙5﹚若AB︰BC﹦5︰6AE﹦7求CF的长
33、已知AD∥BC∠BCD﹦900以AB为直径的⊙O与DC相切于点E,交BC于F﹙1﹚求证AE平分∠BAD﹙2﹚求证AD+BC=AB﹙3﹚求证点E是DC的中点﹙4﹚试判断AD与FC的关系,并证明﹙5﹚求证DC2=4AD·BC﹙6﹚试判断AD,AEAB三条线段之间数量的关系,并证明
34、如图,割线ABC与⊙O相交于BC两点,D为⊙O上一点,AC⊥OE于F,DE交AC于G,∠ADG=∠AGD﹙1﹚求证AD是⊙O的切线﹙2﹚试探究线段ADABAC之间的数量关系,并证明﹙3﹚若AB=2AD=4EG=2求⊙O的半径
35、如图BC为半园O的直径,AD⊥BCEQ\o\acAB=EQ\o\acAFBF和AD交于E﹙1﹚求证AB2=BE·BF﹙2﹚求证=﹙3﹚连接OE试判断OE与MC的关系,并证明
36、△ABC是⊙O的内接三角形,D是EQ\o\acBAC的中点,延长DACB相交于点G﹙1﹚求证AB·AC﹦AD·AG﹙2﹚若AC为⊙O的直径,∠EAG﹦1200BC﹦6求AG的长﹙3﹚在
②的条件下,求证BC2﹦DA·DG
37、已知,⊙o的内接四边形是ABCD,对角线AC与BD互相垂直于点E﹙1﹚连接AO并延长,交⊙o于点F,求证AB·AD﹦AE·AF﹙2﹚若AF与BD相交于点M,求证AB·BF﹦AC·BM﹙3﹚过点O作ON⊥AB于N,试判断ON与CD之间的数量关系,并证明
38、如图,PA为⊙O的切线,PBC为割线,∠APC的平分线交AB于E,交AC于F交⊙O于Q1求证;∠PAE=∠C
(2)求证AE=AF3求证=4求证=内部资料仅供参考9JWKffwvG#tYM*Jg6a*CZ7H$dq8KqqfHVZFedswSyXTy#QA9wkxFyeQ^!djs#XuyUP2kNXp6X4NGpP$vSTT#UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmUE9aQ@Gn8xp$R##849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z8vG#tYM*Jg6a*CZ7H$dq8KqqfHVZFedswSyXTy#QA9wkxFyeQ^!djs#XuyUP2kNXpRWXmAUE9aQ@Gn8xp$R##849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z8vG#tYM*Jg6a*CZ7H$dq8KqqfHVZFedswSyXTy#QA9wkxFyeQ^!djs#XuyUP2kNXpRWXmAUE9aQ@Gn8xp$R##849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmUE9aQ@Gn8xp$R##849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$U*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89Amv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$U*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz84!z89Amv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$U*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9gTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$U*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5ux^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^GjqvadNu##KNMuWFA5uxY7JnD6YWRrWwc^vR9CpbK!zn%Mz849Gx^Gjqv^$UE9wEwZ#Qc@UE%qYp@Eh5pDx2zVkumgTXRm6X4NGpP$vSTT#ksv*3tnGK8!z89AmYWpazadNu##KNMuWFA5ux^Gjqv^$UE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